# TÀI LIỆU TOÁN - WWW.MOLYMPIAD.NET - ĐỀ THI TOÁN

## $show=home 1. Determine all triples of (not necessarily distinct) prome numbers$a,b,c$such that $a(a+1)+b(b+1)=c(c+1).$ 2. In an isosceles right triangle$ABC$, right angle at vertex$A$, let$E$be the midpoint of side$AC$. The line perpendicular to$BE$through$A$meets$BC$at$D$. Prove that$AD=2ED$. 3. Find all positive integers$x,y$and$t$such that$t\leq6$and the following equation is satisfied $x^{2}+y^{2}-4x-2y-7t-2=0.$ 4. Find the greatest integer not exceeding$A$, where $A=\frac{1}{\sqrt{1}}+\frac{1}{\sqrt{2}}+\ldots+\frac{1}{\sqrt{2013}}.$ 5. Let$ABC$be an acute triangle with altitudes$BE$and$CF$. The line segments$FI$and$EJ$are perpendicular to$BC$($I,J$belong to$BC$). Point$K,L$on$AB$,$AC$respectively such that$KI||AC$,$LJ||AB$. Prove that the lines$EI$,$FJ$and$KL$are concurrent. 6. Solve for$x$$\sqrt{13}\sqrt{2x^{2}-x^{4}}+9\sqrt{2x^{2}+x^{4}}=32.$ 7. Given that$x,y$satisfy the conditions $x-y\geq0,\,x+y\geq0,\,\sqrt{\left(\frac{x+y}{2}\right)^{3}}+\sqrt{\left(\frac{x-y}{2}\right)^{3}}=27.$ Find the smallest possible value of$x$. 8. In a triangle$ABC$, let$m_{a},l_{b},l_{c}$and$p$denote the median length from$A$, the lengths of the internal angle bisectors from$B,C$and its semiperimeter. Prove that $m_{a}+l_{b}+l_{c}\leq p\sqrt{3}.$ 9. a) Let$a_{1},a_{2},\ldots,a_{n}$be$n$rational numbers. Prove that if$a_{1}^{m}+a_{2}^{m}+\ldots+a_{n}^{m}$is an integer for all positive integers$m$, then$a_{1},a_{2},\ldots a_{n}$are integer numbers. b) Does the conclusion above remain valid if one only assumes that$a_{1},a_{2},\ldots a_{n}$are real numbers?. 10. Find all continuous funtions$f:\mathbb{R}\to\mathbb{R}$such that the following identity is satisfied for all$x,y$$f(x)+f(y)+2=2f\left(\frac{x+y}{2}\right)+2f\left(\frac{x-y}{2}\right).$ 11. Let$ABC$be a triangle with side lengths$BC=a$,$AC=b$,$AB=c$and median lengths$m_{a},m_{b},m_{c}$drawn from vertex$A,B$and$C$respectively. Prove that for any point$M\begin{align*}\left(\frac{m_{b}}{b}+\frac{m_{c}}{c}\right)\frac{MA}{a}+\left(\frac{m_{c}}{c}+\frac{m_{a}}{a}\right)\frac{MB}{a}+\left(\frac{m_{a}}{a}+\frac{m_{b}}{b}\right)\frac{MC}{c} & \geq3,\\ \left(\frac{b}{m_{b}}+\frac{c}{m_{c}}\right)\frac{MA}{a}+\left(\frac{c}{m_{c}}+\frac{a}{m_{a}}\right)\frac{MB}{a}+\left(\frac{a}{m_{a}}+\frac{b}{m_{b}}\right)\frac{MC}{c} & \geq4.\end{align*} 12. LetABC$be an isosceles triangle at vertex$A$. A circle$\omega$which touches the sides$AB,AC$meets$BC$at$K,L$.$AK$meets$\omega$at$M$. Let$P,Q$be the reflection points of$K$through points$B,C$respectively. Let$O$be the circumcenter of triangle$MPQ$. Prove that$M$,$O$and the center of circle$\omega$are collinear. ##$type=three$c=3$source=random$title=oot$p=1$h=1$m=hide$rm=hide ## Anniversary_$type=three$c=12$title=oot$h=1$m=hide\$rm=hide

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Mathematics & Youth: 2013 Issue 438
2013 Issue 438
Mathematics & Youth